Weak elastic energy of irregular curves

A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our p-energy is defined through a relaxation process, where a suitable p-rotation of inscribed polygons is adopted. The discrete p-rotation we choose has a geometric flavour: a polygon is viewed as an approximation to a smooth curve, and hence its discrete curvature is spread out into a smooth density. For any exponent p greater than 1, the p-energy is finite if and only if the arc-length parametrization of the curve has a second-order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the pth power of the scalar curvature. Finally, a comparison with other definitions of discrete curvature is provided. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.